The Jurkat–Richert theorem is a mathematical theorem in sieve theory. It is a key ingredient in proofs of Chen's theorem on Goldbach's conjecture.[1] It was proved in 1965 by Wolfgang B. Jurkat and Hans-Egon Richert.[2]
This formulation is from Diamond & Halberstam.[3] Other formulations are in Jurkat & Richert, Halberstam & Richert,[4] and Nathanson.
Suppose A is a finite sequence of integers and P is a set of primes. Write Ad for the number of items in A that are divisible by d, and write P(z) for the product of the elements in P that are less than z. Write ω(d) for a multiplicative function such that ω(p)/p is approximately the proportion of elements of A divisible by p, write X for any convenient approximation to |A|, and write the remainder as
rA(d)=\left|Ad\right|-
| \omega(d) | |
| d |
X.
Write S(A,P,z) for the number of items in A that are relatively prime to P(z). Write
V(z)=\prodp\left(1-
| \omega(p) | |
| p |
\right).
Write ν(m) for the number of distinct prime divisors of m. Write F1 and f1 for functions satisfying certain difference differential equations (see Diamond & Halberstam for the definition and properties).
We assume the dimension (sifting density) is 1: that is, there is a constant C such that for 2 ≤ z < w we have
\prodz\left(1-
| \omega(p) | |
| p |
\right)-1\le\left(
| logw | |
| logz |
\right)\left(1+
| C | |
| logz |
\right).
(The book of Diamond & Halberstam extends the theorem to dimensions higher than 1.) Then the Jurkat–Richert theorem states that for any numbers y and z with 2 ≤ z ≤ y ≤ X we have
S(A,P,z)\leXV(z)\left(F1\left(
| logy | |
| logz |
\right)+O\left(
| (loglogy)3/4 | |
| (logy)1/4 |
\right)\right)+\summ|P(z),4\nu(m)\left|rA(m)\right|
and
S(A,P,z)\geXV(z)\left(f1\left(
| logy | |
| logz |
\right)-O\left(
| (loglogy)3/4 | |
| (logy)1/4 |
\right)\right)-\summ|P(z),4\nu(m)\left|rA(m)\right|.